A Moment of Perfect Clarity II: Consequences of Sparse Sets Hard for NP with Respect to Weak Reductions

TL;DR

This paper explores the implications of NP having sparse hard sets under weak reductions, building on recent advances that connect such conditions to potential complexity class collapses.

Contribution

It analyzes the consequences of NP having sparse hard sets under weak reductions, extending prior work on complexity class collapses.

Findings

Sparse NP-hard sets under weak reductions imply class collapses.

Weak reductions are sufficient to derive significant complexity consequences.

The work generalizes previous results to broader classes of reductions.

Abstract

This paper discusses advances, due to the work of Cai, Naik, and Sivakumar and Glasser, in the complexity class collapses that follow if NP has sparse hard sets under reductions weaker than (full) truth-table reductions.